Find each value. If applicable, give an approximation to four dec...

Question

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 1/e^2

Accepted Answer

Hey, everyone. Here we are asked to evaluate the following expression, the natural log of one divided by E raise to the 19th power and approximate our answer to three significant figures. If necessary. Here we have four answer choice options, answer a negative 19 answer B negative 1/19 answer C 6. and answer D 51.6. So here our first step in evaluating this given expression is to expand this expression by recalling a logarithmic identity which states that the natural log of A divided by B is equivalent to the natural log of A minus the natural log of B. And so now we can compare our given expression to the left side of our formula. And we will see that our value for A is one and our value for B is E raised to the 19th power. So now using the right side of our formula, we can expand our expression using our identified values. So we have the natural log of one minus the natural log of E raised to the 19th power. And now we can just simplify this expanded expression. So first, we can recall that the natural log of one is equal to zero. So this first term, the natural log of one is just zero and we are left with negative natural log of E raised to the 19th power. So now we just want to expand this simplified expression further by recalling another logarithmic identity, which states that the natural log of A raised to the power of X is equivalent to X times the natural log of A. So here we can see comparing our formula to our expression that our value for A is E and our value for XX is 19. So now using this formula to expand our expression, we have the expanded expression negative 19 times the natural log of E. And now we can simplify this expression further by now recalling that the natural log of E is equivalent to log base E of E. And this expression is simply equivalent to one. So here we can simplify our expression by noting that the natural log of E is just one. So we actually have negative 19 times one and simplifying we are just left with negative 19. Therefore, we are left with answer choice A where our evaluation comes out to negative 19. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Find each value. If applicable, give an approximation to four decimal places. See Example 5. ln 1/e^2

Key Concepts

Natural Logarithm (ln)

Exponential Function

Properties of Logarithms

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